Game Theory, Game Situations and Rational Expectations: A Dennettian View

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1 Game Theory, Game Situations and Rational Expectations: A Dennettian View Cyril Hédoin University of Reims Champagne-Ardenne (France) This version: 5 February 2016 Abstract: This article provides a theoretical and philosophical analysis of the account of rational expectations in games recently developed by Aumann and Dreze (Aumann and Dreze 2008) on the basis of the correlated equilibrium solution concept. Aumann and Dreze identify a player s rational expectation with his conditional payoff to a correlated equilibrium in a given game situation. This definition depends on the satisfaction of several assumptions in an epistemic game-theoretic framework: the Information Partition Assumption, the Common Prior Assumption and the Common Knowledge of Bayesian Rationality Assumption. I evaluate these three assumptions on the basis of a particular view about intentionality, Daniel Dennett s intentional stance functionalism. Once rational expectations are interpreted along this Dennettian view, one is no longer committed with endowing the players with implausible cognitive abilities. They rather reflect and explain real patterns at the behavioral level. I argue that as an instantiation of externalism in the philosophy of mind, the Dennettian view provides a plausible defense of the Information Partition Assumption and also offers a new though not entirely convincing interpretation of the Common Prior Assumption. However, it fails to provide a satisfactory rationale for the Common Knowledge of Bayesian Rationality Assumption. Keywords: Rational expectations Epistemic game theory Daniel Dennett Correlated Equilibrium Externalism 1. Introduction While discussions about rational expectations are pervasive in macroeconomics, they are surprisingly scarce in the microeconomic context of game theory. However, since macroeconomic variables are obviously a function of the economic agents choices at the microeconomic level, expectations about the former necessarily depend on expectations about the latter. As a consequence, the appraisal of the rational expectation hypothesis must proceed through a critical examination of the status of the concepts of expectations and beliefs in a strategic context where agents must solve coordination problems. This article proposes to Full Professor of Economics, economics and management research center REGARDS. Contact: cyril.hedoin@univ-reims.fr Paper prepared for the 16 th international conference of the Charles Gide Association for the Study of Economic Thought. 1

2 tackle the issue of the status of rational expectations in a game-theoretic framework on the basis of Aumann and Dreze s (2008) formalization of game situations: a game played in a special context and where a player s expectation depends upon the context the situation. Aumann and Dreze identify a player rational expectation with his conditional payoff to a correlated equilibrium in a given game situation. This definition of rational expectations basically relies on three key epistemic assumptions: the Information Partition Assumption (IPA), the Common Prior Assumption (CPA) and the Common Knowledge of Bayesian Rationality Assumption (CKBRA). The paper investigates the ontological and methodological status of the players expectations (and thus, of these three assumptions) and evaluates the relevance of the rational expectations hypothesis on the basis of an externalist view about intentionality that builds on Daniel Dennett s (1987) intentional-stance functionalism and its interpretation within economics by Don Ross (2005). Once rational expectations are interpreted along this Dennettian view, one is no longer committed to endowing the players with implausible cognitive abilities. They rather reflect and explain real patterns at the behavioral level. I argue that as an instantiation of externalism in the philosophy of mind, the Dennettian view provides a plausible defense of the IPA and also offers a new though not entirely convincing interpretation of the CPA. However, it fails to provide a satisfactory rationale for the CKBRA. The article is organized as follows: the second section presents Aumann and Dreze s game theoretic account of rational expectations and provides an explicit epistemic framework for it. In the process, I provide a full description of the three key assumptions of IPA, CPA and CKBRA. The third section characterizes the Dennettian view of intentionality. It briefly surveys the general view called externalism in the philosophy of mind and focuses more specifically on Dennett s intentional stance functionalism, the latter being a peculiar instantiation of the former. The fourth, fifth and sixth sections respectively deal with the IPA, the CPA and the CKBRA on the basis of the Dennettian view. The seventh section concludes. 2. Aumann and Dreze s Game-Theoretic Account of Rational Expectations Aumann and Dreze s article Rational Expectations in Games (Aumann and Dreze 2008) is one of the few attempts to explicitly characterize the rational expectation hypothesis in a game-theoretic framework. It builds on several of Aumann s key contributions in game theory and interactive epistemology, in particular on Aumann (1987) where the solution concept of correlated equilibrium is formally linked to the assumption of Bayesian rationality. Aumann and Dreze s contribution is significant for while the rational expectation hypothesis is pervasive in macroeconomics, its meaning from a microeconomic point of view has rarely been investigated. As a result, even if the hypothesis is formally well-defined in macroeconomic models, it is not clear what it precisely entails in terms of the agents reasoning abilities and knowledge of others reasoning abilities. Since the value of any relevant macroeconomic variable is necessarily a function of the agents behaviors and the latter are partially due to the agents expectations about others behaviors, it follows that the definition of rational expectations with respect to relevant macroeconomic variables 2

3 necessarily depends on our ability to characterize such expectations in terms of interactive epistemology. 1 Cristina Bicchieri (1993) insightfully notes that the rational expectation hypothesis results from the conjunction of two logically independent assumptions regarding the epistemic rationality of the agents. The first ( strong subjective rational belief ) states that the agents use all the relevant information and do not make systematic (i.e. correlated) mistakes while according to the second ( objectively rational belief ) the belief of any agent is correct (i.e. corresponds to the objective probability distribution). However, according to Bicchieri (1993, 25), the rational expectation hypothesis gives no account of how this coincidence comes about, as there is no plausible theory of how the agents learn to be epistemically rational in the sense specified by [the objectively rational belief assumption]. As I explain below, even though Aumann and Dreze s account does not offer any hint regarding how the agents may learn the objective probability distribution, it provides a clear-cut statement of the conjunction of the two epistemic assumptions underlined by Bicchieri but also shows that this conjunction is not sufficient in itself. I start by sketching Aumann and Dreze s account that I then reformulate in an explicit epistemic framework. Aumann and Dreze s main purpose is to characterize an agent s rational expectation in terms of the payoff that he can rationally expect to have in a particular game situation, i.e. a game played in a specific context where a player s expectation depend upon the context the situation (Aumann and Dreze 2008, 72). More precisely, they propose to define a player s rational expectation as his conditional payoff to a correlated equilibrium in a given game situation. Consider a generic strategic interaction that we describe through some game G: < N, {Si, ui}i N > where, as usual, N is a set of n 2 players i = (1,, n), Si is the finite set of player i s pure strategies and ui: S i s utility function mapping any strategy profile belonging to S = isi onto some real number. A game situation corresponds to any particular instantiation of G where each player possesses some specific (private) information about anything that is relevant from his point of view, in particular the other players choices and beliefs. Let characterize such a game situation as an epistemic game w: < G, I, w > where I is an information structure or a broad theory of G that specifies what each player knows and believes about others and how he reasons on the basis of this information. A correlated equilibrium in G corresponds to a correlated distribution of strategy profiles defined by some variable f(.) such that each player maximizes his expected utility in each strategy profile given the information available to him. A player s conditional payoff to a correlated equilibrium is defined as the player s expected utility at an information set, i.e. what he expects to gain by implementing his strategy in the corresponding strategy profile given the information available to him, assuming that others play along the correlated equilibrium. Therefore, a player s rational expectation corresponds to what he can expect to gain if he implements the strategy constitutive of a given correlated equilibrium defined by the function f(.), conditional on the information he has about his own behavior (possibly among other things). 1 Informally, the rational expectation hypothesis is generally stated by ambiguous sentences like the individuals know the relevant macroeconomic theory, the agents correctly predict the value of macroeconomic variables or even more loosely one cannot be fooled systematically. Of course, the hypothesis has a quite clear formal expression, namely that for any agent i and for any macroeconomic variable X, i s expectation at time t of the value of X at time t+1 corresponds to the actual value of X at t+1 on average, i.e. E(X t+1 ) = X t where 0 is a random error variable of mean 0 and E the expectation operator. 3

4 As a simple illustration, consider the following hawk-dove game: Figure 1 H D H 0 ; 0 5 ; 1 D 1 ; 5 4 ; 4 This game has two Nash equilibria in pure-strategy, yielding (1, 5) and (5, 1) and one in mixed-strategy where each player plays H with probability ½, thus yielding (5/2, 5/2). However, there are also a wealth of correlated equilibria, such as the one defined by the following probability distribution over the strategy profiles: f([d, D]) = 1/3, f([h, D]) = 1/3, f([d, H]) = 1/3, yielding (3, 3). This distribution constitutes a correlated equilibrium as it is not difficult to find that it is optimal for a Bayesian rational player to implement the strategy corresponding to each strategy profile given the conditional probabilities derived on the basis of f(.). 2 According to Aumann and Dreze s definition, a player s rational expectation when he plays H is then 5, while it is 5/2 when he plays D. 3 There are several ways through which we can generalize this example and formalize the notion of game situation. This depends on how we represent the information structure I in the epistemic game w. Aumann and Dreze (2008) characterize the information structure in terms of a type space T. Each player i is endowed with a finite set of types Ti where each type ti specifies a) the player s choice and b) the player s belief regarding the type of the other players. As each player s type defines a first-order belief over others types, it is easy to see that we can associate to any type profile (t1, t2,, tn) an infinite belief hierarchy that specifies all the higher-order beliefs of the players (i.e. what Row believes about what Column believes about what he will play). A more natural but formally equivalent representation is in terms of a state space. A state (or possible world) w is an exhaustive description of everything that is relevant for the players and the modeler. In a game-theoretic context, it specifies in particular the players choices, their beliefs about others choices, their beliefs about others beliefs and so on. Basically, at a given state, no uncertainty remains relatively to the value of any relevant variable. 4 Formally, one can define a state w simply as a specific 2 Consider Row player (the same reasoning applies for Column player). Obviously, it is optimal for him to play H in the profile (H, D) as in this case he knows with certainty that Column plays D. When Row plays D, he assigns a conditional probability of ½ to Column playing H and of ½ to column playing D. Then, by playing D as indicated, his expected gain is 5/2. If he plays H instead, his expected gain is also 5/2, thus he has no incentive to deviate. Therefore, the probability distribution defines a (weak) correlated equilibrium. 3 An interesting result discussed by Aumann and Dreze is that the players rational expectations may be mutually inconsistent in the sense that the resulting payoff profile is infeasible, i.e. it is outside the convex hull of the possible payoff vectors. This shows that rational expectations in a strategic context do not entail efficiency or even consistency when players have differential information. 4 The state characterization (like the type characterization) corresponds to a semantic model in the sense that it consists to assigning a truth value to a list of propositions. A state is thus a list of propositions (about the players behavior, about their beliefs, and so on) that are true. Such a semantic framework has a syntactic counterpart which is built from a language consisting in atomic propositions, logical connectives and modal operators. By combining this language with a set of axioms, we can derive a set of theorems. A syntax is sound and complete with respect to a given semantic model (or class of models) if all theorems in the syntax are valid in the semantic model (i.e. true at every states) and all valid propositions in the semantic model can be proved as theorems in the syntax respectively. In economics, it is usual to left the syntax implicit. See however Aumann (1999) and Bacharach (1994) for a discussion of the relationship between syntax and semantics in game theory. 4

5 type profile (t1, t2,, tn), i.e. = T1 x T2 x x Tn, which means that we can also ascribe to each state a strategy profile and a belief hierarchy. A second component of the information structure is a vector of prior probabilities functions {pi(.)}i N defined over the state space. Formally, the function pi(.) assigns to each state w a prior probability on the basis of which each player updates his belief conditioning on the information received. As I explain below, the proper economic interpretation for the prior function is unclear but it still plays a crucial role in the derivation of Aumann and Dreze s results. The third component of the information structure consists in a vector of accessibility relations {Ri}i N which states for each state w and for each player i which are the states w that are accessible, denoted as wriw (i.e. w is accessible for i from w). The accessibility relation can be interpreted in several ways; in the epistemic context which is relevant here, the appropriate interpretation is in terms of epistemic possibility: wriw if and only if when at state w, i considers possible to be at w. I denote Ri(.) the corresponding possibility operator where Ri(w) is the set of states w that are accessible for i at w. The players posterior probabilities pi,w(.) are accordingly defined over Ri(.) using Bayes rule. The tuple I: <, {pi, Ri}i N > is thus the broad theory of game G, the complete description of whatever may happen or could have happened in G. Finally, denote w the actual state, i.e. the way the game is actually played and what the players actually know and believe. A game situation then is formalized as an epistemic game w: < G, <, {pi, Ri}i N >, w >>. As explained above, Aumann and Dreze characterize a player s rational expectation as his conditional payoff to a correlated equilibrium in a given game situation w based on game G. I denote s(w) = (s1,, sn) the strategy profile that is implemented at w and E[ui(s(w) Ri(w)] player i s expected payoff when he plays his part in s(w) conditional on his information at w (defined by his possibility operator Ri and his posterior probability pi,w). Now, the broad theory I implements a correlated equilibrium in G only if for all w, all players i and any strategy s i si, (1) E[ui(s(w)) Ri(w)] E[ui(si, s-i) Ri(w)], with s-i = (s1,, si-1, si+1,, sn). Then, in any game situation w, a player i s rational expectation corresponds to E[ui(s(w) Ri(w)], i.e. his expected payoff at the actual world w. This definition of rational expectations depends on several assumptions that are related to the players rationality, both practical and epistemic. Actually, these assumptions follow from a theorem that has been established by Aumann (1987) regarding the sufficient conditions for a correlated equilibrium to be played. 5 For the rest of the paper, I will call it Aumann s Theorem : Aumann s Theorem Consider a broad theory I of game G such that all players have a common prior p = p1 = = pn. Then, the probability distribution of strategy profiles s(w) is a correlated distribution f(.) such that f(s) = p(w). Moreover, if all players are Bayesian rational at all states w, then f(.) corresponds to a correlated equilibrium in G. Proof See Aumann (1987) and Gintis (2009, 138-9). 5 See also Gintis (2009, 138-9). 5

6 Basically, since each state w specifies a unique strategy profile s(w) to be implemented, it follows that the probability distribution of these profiles is defined by the common prior p. In other words, the common prior p in I implements the correlated equilibrium f(.) in G. Aumann s Theorem makes two explicit requirements: first, there must be a common prior over the state space; second, all the players must be Bayesian rational at all states of the world: CPA (Common Prior Assumption): i N: pi = p. CKBRA (Common Knowledge of Bayesian Rationality): i N, w : E[ui(s(w)) Ri(w)] E[ui(si, s-i) Ri(w)]. These two assumptions are mathematically straightforward. CPA states that all the players share the same ex ante belief (i.e. before receiving any private or public information) over what can happen in G. Another way to state this assumption is that there is an (possibly tacit) agreement among the players regarding the fundamental features of the social world. CKBRA indicates that the players are Bayesian rational, that everyone knows that, that everyone knows that everyone knows that, and so on ad infinitum. 6 Aumann s Theorem also relies on a third, implicit assumption according to which each player must have an information partition Ii over. This is derived from the properties of the possibility operators Ri: IPA (Information Partition Assumption): The possibility operators Ri have the following properties, 7 a) i N, w : w Ri(w). b) i N, w, w : if Ri(w) Ri(w ), then Ri(w) Ri(w ) =. Property (a) states that a player always considers the actual state of the world to be possible. Property (b) is a direct statement that each player has a partition over : if at w, i considers w as possible but w as impossible, then at w he cannot also consider w possible. Therefore, is divided into cells with no intersection. 8 The foundations of rational expectations in games are thus now explicit: the players must have an information partition and a common prior, and Bayesian rationality must be common knowledge. Though they are somewhat standard in economics (especially in information economics), these assumptions are also all controversial. I will present and discuss a rationale 6 A player is Bayesian rational if he maximizes his expected utility given his subjective beliefs and if he uses Bayes law to update his beliefs conditional on some information. 7 IPA could also be stated in terms of the properties of the accessibility relation R i from which the possibility operator is derived. In this case, R i must be reflexive (wr iw), transitive (if wr iw and w R iw, then wr iw ) and Euclidean (if wr iw and wr iw, then w R iw ). 8 Formally, property (b) is sufficient for the players to have a partition I i over. However, Aumann s Theorem as well as Aumann and Dreze s account are couched in terms of both belief and knowledge, i.e. probability 1 beliefs that are true. Property (a) is then required to make sure that what the player believes with probability 1 is indeed true. We could dispense with this requirement if instead we choose to frame the whole discussion in terms of beliefs uniquely. I will not go into these subtleties as they do not affect the main points of my argument. 6

7 for them based on a Dennettian view of intentionality in sections 4, 5 and 6. Before, I characterize this Dennettian view and explain why it is relevant here. 3. A Dennettian View of Intentionality: Externalism and Intentional-Stance Functionalism It is important to emphasize that Aumann and Dreze s do argue neither for the empirical plausibility nor for the theoretical relevance of rational expectations. Their goal is purely formal: to provide a mathematical characterization of rational expectations in explicitly strategic contexts. I shall argue however that this mathematical characterization could potentially enhance both the empirical plausibility and the theoretical relevance of the rational expectation hypothesis, provided we adopt a specific view about intentionality that I call the Dennettian view. The rational expectation hypothesis has been disputed both at the theoretical and the empirical levels. I have already mentioned one objection made by Bicchieri (1993): the rational expectation hypothesis does not provide any argument for the assumption that the agents subjective beliefs always match the objective probability distribution of events in the world. There are clearly grounds to doubt this assumption even for exogenous events that do not depend on the agents choices and beliefs. Doubts are only strengthened when we consider events whose probability distribution is endogenously determined. Moreover, the rational expectation hypothesis builds on the strong assumption that errors tend to cancel out, i.e. agents may make mistakes (choices that do not maximize expected utility) but these mistakes are randomly distributed such that they do not play any role at the aggregate level. This is dubious from a purely empirical point of view, at least until we show that there are institutional structures that have the property to generate some form of ecological rationality (Smith 2009). One may argue that Aumann and Dreze s characterization of rational expectations in games suffers from the same problem. Consider this line of argument: according to Aumann and Dreze, from her point of view in some game situation w, Ann rationally expects to gain E[uAnn(s(w)) RAnn(w)]. Such expectation however depends on the fact that a specific correlated equilibrium corresponding to a common prior p is implemented. But there are typically many correlated equilibria in a game. 9 Why should Ann have any particular reason to expect that this particular equilibrium will be played and not any other possible one? Why should she even expect that any correlated equilibrium will be played? We could provide many specific answers to these questions: maybe Ann has previously agreed with Bob that the strategy profile to be implemented should be a function of the result of a coin toss or of the weather. Maybe Ann has observed that in the past, people s behaviors were correlated to some external signal (e.g. in most places on the planet, people seem to stop at red traffic lights but not at green ones). Or maybe this is due to a purely genetic disposition that has programmed Ann to identify some asymmetries in an interaction and to adopt a particular behavior on this basis (Skyrms 1996, chap. 4). These are all proximate 9 Consider the game described by Fig. 1 above. It is easy to see that there are an infinity of correlated equilibria in this game because any probability distribution of the two Nash equilibria [H, D] and [D, H] is a correlated equilibrium. More generally, since any Nash equilibrium is a correlated equilibrium while the reverse is not true, the number of correlated equilibria in a game is always even or greater than the number of Nash equilibria. 7

8 explanations. But the only ultimate and general one has already been provided: Ann s rational expectation is grounded on the fact that the IPA, CPA and CKBRA are all satisfied. If one wants to make sense of rational expectations in games both from a theoretical and empirical point of view, then the task is to provide a rationale to these three assumptions. There are many arguments to reject all of them that I will examine in due course. If these arguments are valid, then we should not expect people to empirically have rational expectations but we should also be skeptical regarding the methodological and the theoretical value of the rational expectation hypothesis. I shall argue that the Dennettian view of intentionality provides maybe the best defense of the three assumptions even if ultimately there are reasons to consider that it is not entirely successful. Indeed, expectations and beliefs are what philosophers of mind call intentional states. 10 The ontological status and the methodological implications of rational expectations thus depend on one s views about intentionality and intentional states. As argued by Ross [(2005); (2014)] in the case of revealed preference theory, Daniel Dennett s (1987) intentional-stance functionalism provides an interesting account of the nature of intentional states paving the way for a reinterpretation of economic theory. There are reasons to think that the same may be true in the case of rational expectations as characterized above. Before describing Dennett s account further, it might be helpful to consider a key distinction in the philosophy of mind between internalism and externalism, Dennett s intentional-stance functionalism being a peculiar instance of the latter. The distinction concerns the nature and the origin of the semantic content of intentional states. Any intentional state (also called intentional attitude) is of the general form F(x) where F is the type or mode of state (belief, desire, intention ) and x the propositional content, i.e. what the state is about. For instance, the fact that I believe that the Golden State Warriors won the NBA Finals in 2015 is an intentional state F(x) where F corresponds to the type belief (a cognitive attitude with a mind-to-world direction of fit) and x is the object of the belief, i.e. that the Golden State Warriors won the NBA Finals in A key feature of such intentional attitudes is their aboutness : they are about something that is distinct and in some way external to the person who holds the attitude. In other words, intentional attitudes represent non-mental properties or states of affairs. The propositional content of an intentional state also represents the conditions of satisfaction of this state: the fit between the intentional state and the actual state of affairs is achieved if and only if the latter matches with the propositional content. In the case of a belief, a belief F that x is true if and only if x actually holds as a state of affairs. The debate between internalism and externalism about the propositional content of intentional states concerns the way the meaning of the propositional content x in F(x) is determined. Internalism holds that the meaning of x is intrinsic to the entity (the person) that holds the state F(x). Alternatively, we might say that the meaning of x supervenes on the intrinsic properties of the entity. Externalism holds precisely the converse: the meaning and even the very existence of the propositional content is partially determined by the relationship of the entity with its environment. In other words, the semantic content of intentional states depends on the whole situation in which these states are embedded. 10 The same is true for preferences, assuming that they represent desires and wants. 8

9 Though the debate is far from having been completely settled, externalism has acquired a dominant position in the philosophy of mind and in the cognitive sciences. 11 For some kinds of intentional states, the case for externalism is almost straightforward (Lau and Deutsch 2014). This is the case for instance regarding knowledge about the external world: I can know that the Warriors won the 2015 NBA Finals only if indeed it is the case that Warriors won the 2015 NBA Finals. The disagreement between internalists and externalists rather holds for intentional states that do not imply a veridical content. Since this paper is concerned with the status of (rational) expectations, it is especially important to state what externalism for beliefs implies. Consider this modified version of Hilary Putnam s (1975) Twin Earth example : suppose that somewhere in the Universe (or in one among many universes if one holds that there are parallel worlds) there is a planet almost identical to ours that we call Twin Earth. Suppose that the only difference between Earth and Twin Earth is that on the latter people do not play basketball exactly in the same way as we play it on Earth (we may assume for instance that tackles are legal and that you can walk with the ball without making it rebounding). Suppose finally that the NBA Finals take place each year on Twin Earth to determine which team will be the champion in the same way as on Earth. Now, someone committed to externalism about beliefs will hold that the content of any belief related to the game of basketball (such as which team will win the NBA Finals this year) has no intrinsic meaning but instead depends on whether the entity having this belief is on Earth or on Twin Earth. For instance, Bob and Twin-Bob (Bob s counterpart on Twin Earth) may hold exactly identical beliefs about states of affairs related to basketball, e.g. both may believe that the Warriors will win the 2016 NBA Finals. However, according to externalism, Bob s and Twin- Bob s beliefs are not the same because they refer to qualitatively different things. Meaning is not in the head but instead is determined by the whole relationship between the believer and his environment. I do not intend here to evaluate the virtues of externalism with respect to beliefs against various forms of internalism. As I said above, externalism has currently the upper-hand among philosophers of mind and cognitive scientists (at least those concerned with such philosophical issues). This fact is by itself sufficient to justify that we investigate the implications of externalism for economics. The point is thus to reinterpret the meaning of rational expectations in games from an externalist point of view and to determine whether this reinterpretation reinforces the rational expectation hypothesis, from a methodological, a theoretical and possibly an empirical perspectives. At the same time, it is clear that externalism is not a strongly committing doctrine: we can all be externalists regarding the content of intentional states and yet defend a great diversity of positions about a range of issues such as the mind-body problem and the nature of consciousness, mental causation or the problem of intentionality. Dennett s intentional-stance functionalism is a particular instantiation of externalism whose significance for economics has already been argued by Ross (2005). As Ross convincingly shows (and as Dennett has punctually remarked), there is 11 Burge (1986) and Putnam (1975) provide two well-known and early defenses of externalism about the content of intentional states, building on a related but different form of externalism, semantic externalism. Crane (2015) and Searle (2004) present introductory remarks about the externalism/internalism debate, both endorsing internalism. Dennett explicitly endorses externalism at several places, see especially his article True Believers: The Intentional Strategy and Why it Works, reprinted in Dennett (1987, 13-35). 9

10 a deep affinity between Dennett s views about what he calls intentional systems and the way agents are formalized in economics, especially in decision theory and game theory. 12 Dennett s account is not easy to recapitulate because it has several ramifications and subtleties. I will confine myself to the way Dennett interprets intentional states and particularly beliefs. As its name indicates, intentional-stance functionalism is a specific form of functionalism. The latter is itself a broad view according to which mental events (including intentional states) supervene on physical events and that the relationship between the two kinds of events is a functional one: a given physical event (or state) E is a mental event (a state) in virtue of its function in the overall behavior of the entity. An interesting implication of functionalism is the relative independence between the characteristics of the physical structure of the entity (the hardware) and its functional activity (the software): how a particular functional activity is implemented is secondary and in principle any functional activity can be implemented by any physical structure. 13 Functionalism takes several shapes and has for instance led some scholars to lean toward eliminativism, i.e. the doctrine holding that the categories of folk psychology (beliefs, desires) correspond to an erroneous theory of mind and that scientific discussions should refrain from using them. Though it has been frequently associated to such an eliminativist endeavor, Dennett s intentional stance functionalism firmly rejects eliminativism. 14 Quite the contrary, Dennett s account can be seen as an attempt to rehabilitate the categories of folk psychology not only from an epistemological point of view (i.e. as useful concepts to predict people s behavior) but also from an ontological point of view. The nature of intentional states such as beliefs is deeply related to the existence of what Dennett calls intentional systems : What is it to be a true believer is to be an intentional system, a system whose behavior is reliably and voluminously predictable via the intentional strategy (Dennett 1987, 15). The intentional strategy consists precisely in predicting one s behavior through the attribution of intentional states. Thus, reformulating a) An entity E has belief B(E) only if E is an intentional system S. b) E is an intentional system S only if its behavior is reliably predictable via the intentional strategy I. c) The intentional strategy I consists in predicting E s behavior by attributing to E some belief B(E) (possibly among other intentional states). d) Therefore: An entity E has belief B(E) only if E s behavior can be predicted by attributing to E belief B(E) (possibly among other intentional states). The intentional strategy for predicting someone s behavior is what Dennett also calls the intentional stance. It corresponds to one of the three available epistemological postures (along with the physical stance and the design stance) to predict and explain a system behavior. 12 Dennett has not written anything specific about economics and its methodology. However, he notes that his intentional system theory overlaps with branches of economics and especially game theory at several places. See for instance his article Three Kinds of Intentional Psychology reprinted in Dennett (1987, 58). 13 This point of view is held in particular by what is sometimes called computer functionalism or strong artificial intelligence (Searle 2004). The Church-Turing thesis and the related notion of Turing machines are at the core of computer functionalism as the former holds that any computable function is computable by a universal Turing machine (the Turing machine that computes all the functions computed by all Turing machines). If one assumes that the mind is a computable function (a premise which is of course debatable), then the Church-Turing thesis strongly indicates that strong artificial intelligence is possible. 14 For an explicit departure from eliminativism, see for instance Dennett (1987, ). 10

11 Taking the intentional stance toward someone (or something) consists in explaining and predicting the latter s behavior through the attribution of intentional states. The intentional stance has an apparently strong instrumentalist flavor that corresponds to what can be called the Dennettian method for behavioral explanation (Ross 2002, 154) or methodological intentional-stance functionalism (Ross 2005). However, a purely instrumentalist reading of Dennett s account would indicate that belief attribution is merely based on a falsifiable theory of the mind and that we should give it up provided we are able to show that this theory is false or unnecessary (with respect to some parsimony criterion). This is precisely the eliminativists position that Dennett rejects. Consider indeed hypothetical Martians who are observing Humans and are trying to predict our future on the basis of superhuman abilities making them equivalent to Laplacean superphysicists: Our imagined Martians might be able to predict the future of the human race by Laplacean methods, but if they did not also see us as intentional systems, they would be missing something perfectly objective: the patterns in human behavior that are describable from the intentional stance, and only from that stance, and that support particular generalizations and predictions (Dennett 1987, 25, emphasis in original). Dennett s point is that the intentional stance is not merely instrumental; it is the only way to observe real behavioral patterns. In this sense, the intentional stance is not a theory that might be proved to be wrong as eliminativists would hold. It is constitutive of real patterns that we cannot characterize but in terms of beliefs, desires and other kinds of mental states. According to Dennett s ontological intentional-stance functionalism (Ross 2005), there is thus nothing more in the fact that the entity E has the belief that than the fact that E s behavior can be interpreted and predicted (by E itself or others) from the intentional stance through the ascription to E of the belief that. This is a form of realism, though a mild one since in many cases one s mental states will be partially indeterminate from the intentional stance (Dennett 1991). 15 We are now in the position to characterize the Dennettian view of intentionality, especially of beliefs. On this view the semantic content of beliefs and any other intentional states is not intrinsic to the entity holding them (Externalism). It depends on the functional relationship between the entity and its environment (Functionalism). This semantic content is fixed by adopting the intentional stance: this entity has the beliefs and the other intentional states that make its behavior the most understandable and predictable, assuming that the entity is endowed with at least a minimal form of rationality (i.e. we attribute to the entity the beliefs and desires it ought to have given its behavior and the environmental context) (Intentional Strategy). Moreover, this is all there is to have a belief though in many cases the precise content of the belief will be indeterminate (Mild Realism). For the rest of the paper, the Dennettian view will denote the conjunction of Externalism, Functionalism, Intentional Strategy and Mild Realism. 15 This indetermination has a strong formal similarity with Quine s radical translation problem, as Dennett notes at several places. 11

12 4. Externalism and the Information Partition Assumption I now examine the IPA, CPA and CKBRA on the basis of the Dennettian view presented above, starting with the IPA. Recall that the IPA states that each player i has an information partition Ii over the state space. To have an information partition means that each state w belongs to one and only one set defined by the possibility operator Ri(.). As a consequence, ignoring the limit case where Ri(w) = for all w, at any w the player i has some information such that the state space can be divided into the states w that he knows are impossible (i.e. w Ri(w)) on the one hand and the set of states w he knows are possible (i.e. w Ri(w)) on the other hand. Moreover, we have assumed that w Ri(w) which means that i necessarily knows at w that w is possible. It follows that the partition Ii defines i s knowledge in any given game situation. To see this point, consider the following definitions. Definition 1 An event E is a subset of states such that a given proposition is true at all w E. Definition 2 The event that player i knows an event E is denoted KiE and is defined as follows: KiE = {w Ri(w) E}. According to Definition 2, i knows E at w if and only if all the states i considers as possible belong to E. Note that KiE is itself an event since it corresponds to set of states where the corresponding proposition is true. Accordingly, Ki is called a knowledge operator. On the basis of these definitions and of the IPA, it can be shown that the knowledge operator satisfies the following axioms for any events E and F: 16 (K1) (K2) (K3) (K4) (K5) Ki = E F KiE KiF KiE E KiE = KiKiE KiE Ki KiE The first two axioms are axioms of (logical) omniscience as K1 states that one knows everything that is necessarily true and K2 that one necessarily knows the logical consequences of what he knows. K3 is generally known as the truth axiom and is constitutive of the definition of knowledge: one can only know things that are true. K4 and K5 are sometimes called axioms of transparency (or of positive introspection) and of wisdom (or of negative introspection) respectively. The former states that one knows that he knows and the latter that when one does not know something he knows this (or equivalently, that if one does not know that he does not know something, then he knows this something). It is now largely established that these five axioms are the semantic equivalent to the well-known S5 modal logic at the syntactic level (e.g. Stalnaker (2006)). From this point of view, even though K1 and K2 lead to the problem of logical omniscience, they are generally considered as quite standard by logicians. 17 Axioms K3, K4 and K5 are more disputable, particularly in the context of 16 See for instance Binmore (2007) or Gintis (2009) for simple derivations. 17 K1 and K2 alone correspond to the modal logic K which is taken as the generic system of modal logic. Logical omniscience may be regarded as problematic as it endows agents with excessively strong cognitive and 12

13 economics. K3 implies that what one believes with certainty is necessarily true, thus forbidding the possibility that beliefs with probability one are false. At the same time, K3 is required as soon as one wants to deal with problems of information in terms of knowledge as it is generally the case in economics. K4 and especially K5 cannot be defended on this basis: they arguably define very strong requirements of epistemic rationality that seem hard to ground on a reasonable psychological and/or economic rationale. Still, like K3, they directly follow in Aumann and Dreze s account from the fact that it is implicitly assumed that each player knows his type ti. Since a state w corresponds to a profile of types (t1,, tn), it follows easily from this that if any player i knows the event that he is of type ti then he is indeed of type ti (K3). Moreover, since to be of type ti implies to know his type, knowing his type implies to know this (K4). Finally, since i knows that he is of type ti, he knows when his type does not know something (K5). IPA is thus implied by the type-space approach used by Aumann and Dreze. I shall argue that the Dennettian view provides a convincing case for the IPA in a gametheoretic context. This claim can be supported by several arguments. First, a case can be made for the fact that the axioms of S5 system in epistemic logic and thus the IPA are innocuous in a small set of specific contexts (Lismont and Mongin 1994). For instance, it has been argued that the use of the so-called Kripke structures is justified for the study of systems with distributed knowledge, i.e. systems where distributed information is computed by parallel processors (Halpern and Moses 1990). Relatedly and more significantly, the use of information partitions seems to follow naturally from an external view of knowledge, i.e. knowledge as ascribed by the scientist rather than computed by the agent (Lismont and Mongin 1994, 91). This external view of knowledge is clearly deeply related to externalism about intentional states. As I have noted above, externalism about knowledge is almost uncontroversial because knowledge implies a veridical content: one can only know something that is true and truthfulness implies a relationship between a mental state and some external state of affairs. In this sense, the ascription of knowledge to an agent (possibly by the agent himself) implies a reference to the environment. On this view, there is nothing like an inner state of knowledge and all the axioms of the S5 modal logic (including the axiom of wisdom 18 ) seem to be defensible on this basis. However, this argument clearly depends on a definition of knowledge as true belief that may be rejected for independent reasons. A second argument still relies on an externalist view of knowledge but also more specifically on Dennett s intentional-stance functionalism. The uneasiness with the axioms of the S5 modal logic in an epistemic context stems from the fact that the computation of all the required information is cognitively out of reach for normally rational humans. In particular, the omniscience axioms (K1 and K2) appear to be particularly strong as they endow the agents with perfect logical abilities. The axioms of transparency and wisdom (K4 and K5) assume that the agents know what they would know and not know in any (counterfactual) computational abilities. See Bacharach (1994), Halpern and Pucella (2011) and Sillari (2008), among others, for discussions of this problem in economics, computational science and philosophy respectively. 18 If the proposition that Ann does not know that is true, then Ann can potentially have the knowledge that she does not know that. Now, from an external point of view, the ascription of such a knowledge is permissible as long as nothing in Ann s relationship with the external world indicates (through her behavior for instance) that she is ignorant of her ignorance. The latter is a possibility of course, but it is in principle always possible to change our description of this relationship (formally, by redefining the state space and the corresponding information partitions) such that Ann knows her ignorance. This point is related to Dennett s remark that intentional states may be indeterminate and Quine s radical translation problem. 13

14 state of the world. In other words, they depend on the fact that the agents are able to compute all the information, including the information that is not actually available. Such a computational load seems to be intractable given the cognitive capacities of any normally intelligent human. There are two complementary answers to this objection building on externalism and intentional-stance functionalism. On the one hand, on an externalist reading of knowledge and other intentional states, we need not assume that the agents really make all the computations that are reflected in a given epistemic model. That is, we do not require that the agents are able to make explicit their reasoning and the knowledge on which it is based, or that they have some inner, privileged and/or conscious access to the underlying computational processes. The distinction between explicit and implicit knowledge widely used in the literature on the so-called awareness structures captures this point. 19 Under this terminology, an agent explicitly knows that if and only if he implicitly knows that and he is aware of, where the latter could here mean being conscious of. The K1-K5 axioms are clearly problematic in terms of explicit knowledge for the reasons just stated. However, there is a priori no reason to reject these axioms once it is acknowledged that they are about implicit knowledge. From the standpoint of (computer) functionalism, the use of knowledge in the sense of implicit knowledge is quite natural as the computation associated to the intentional state ascribed to the agent is not an inner one. On the other hand, Dennett s intentional stance functionalism provides a rationale to start from the assumption of perfect rationality : That is, one starts with the assumption that people believe all the implications of their beliefs and believe no contradictory pairs of beliefs (Dennett 1987, 21). The first part of this assumption corresponds to axiom K2, while the second corresponds to an axiom that is weaker than the truth axiom K3. 20 Moreover, on the intentional strategy, the attribution to an intentional system of possibly false beliefs necessarily requires a special genealogy, which will be seen to consist in the main in true beliefs An implication of the intentional strategy, then, is that true believers mainly believe truths (Dennett 1987, 18-9). Thus, Dennett s intentional strategy clearly recommends to ascribe beliefs (either true or false) through the ascription of what we can call a knowledge base, i.e. a set of basic beliefs that are true. This clearly supports the truth axiom K3. Note that these basic beliefs are supported by the agents possibility operators Ri as one s knowledge is formally represented by the set Ri(w) for each state w. This requires that these sets do not intersect which is guaranteed by axiom K4 (Bacharach 1993). Indeed, the contrary would imply that one can know contradictions, which is impossible. 21 It is less clear that the intentional strategy provides an independent support for the axiom of wisdom K5. Without it, the resulting system of axioms (known as S4) allows for the possibility operators Ri to define a topology instead of a partition and Aumann s Theorem no longer applies. However, we have already seen that K5 is not unreasonable on an externalist 19 For a survey of awareness models in computational science, philosophy and economics, see Sillari (2008). 20 Formally, K ie K i E. This axiom (generally denoted axiom D in the modal logic literature) applies to an epistemic logic for beliefs instead of knowledge. When D is substituted for K3, the resulting system of modal logic is called KD Suppose that an agent i has the following information pattern over the space = (1, 2, 3): Ri(1) = (1, 2) and Ri(2) = Ri(3) = (2, 3). Therefore, the two sets intersect at world 2. However, this pattern is impossible if axiom K4 is satisfied: assume that the actual world is 1; then, i knows that 2 is possible but that 3 is impossible, but at 2 he knows that 3 is possible. Since K4 indicates that i knows what he knows and since one cannot know contradictions (by the axioms of propositional logic and K3), he cannot know that he knows that 3 is both possible and impossible. 14